Delphine LAUTIER and Alain GALLI Delphine LAUTIER(*) is ...

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The remainder of the paper is organized as follows. Section II presents the term structure models of commodity prices and explains why their use necessitates resorting to the Kalman filters. Section III explains how to apply the simple and the extended Kalman filters to a well-known model developed by Schwartz in 1997. Relying on the model performances, section IV compares the two filters and discusses some practical implementation problems. Section V concludes. II. THE TERM STRUCTURE MODELS OF COMMODITY PRICES In this section, after describing some general features characterizing the term structure models of commodity prices, we present the model used for the comparison between the simple and the extended Kalman filters: Schwartz’s model. General presentation The term structure of commodity futures prices describes the relationships between the spot price and futures prices for different delivery dates. So it synthesizes all the information available in the market. Several term structure models have been proposed in the literature. Their objective is firstly to reproduce the observed futures prices as accurately as possible, and secondly to extend the curve for very long maturities, even for delivery dates which are not available in the market. Term structure models borrow from the contingent claim analysis developed in a partial equilibrium framework for options and interest rates models. Relying on arbitrage reasoning, the development of a term structure model of commodity prices follows three successive steps: identification of the state variables, specification of their dynamics and extraction of the futures prices values from a differential valuation equation. When only one state variable is used to explain the futures prices behaviour, as is the case, for example, in Brennan and Schwartz’s model (1985), this single factor is the spot price. Recognizing the limits of such a formulation, several models based upon two 4
state variables have been proposed (Schwartz, 1997; Hilliard and Reis, 1998; Lautier, 2000). In that case, the second factor is the convenience yield, which can be briefly defined as the comfort associated with the possession of physical stocks (Brennan, 1958). The introduction of a second state variable allows for richer shapes of curves and volatility structures. This improvement is however costly because the models are naturally more complicated. The difficulty arises from the increasing number of parameters and from the non-observable nature of the state variables. In fact, there are usually no empirical data for these two variables because there is generally a lack of reliable time series for the spot price 5 , and convenience yield is not a traded asset. Therefore, there is a need for a method like the Kalman filter. Schwartz’s model Schwartz’s model (1997) is a well-known term structure model of commodity prices. Three reasons lead to choose it. Firstly, it performs well. Secondly, it has an analytical solution, which simplifies the application of the Kalman filters. Thirdly, it allows for the use of a simple Kalman filter, provided some precautions are taken. Schwartz’s model supposes that the spot price S and the convenience yield C can explain the behavior of the futures prices F. The dynamics of these state variables is: ⎧dS = ( µ − C) Sdt + σ S Sdz ⎨ ⎩dC = [ k( α − C) ] dt + σ Cdz with: κ, σS, σC >0 where: - µ is the drift of the spot price, - σ S is the volatility of the spot price, - dzS is an increment to a standard Brownian motion associated with S, - α is the long run mean of the convenience yield, - κ is the speed of adjustment of the convenience yield, - σ C is the convenience yield volatility, - dzC is an increment to a standard Brownian motion associated with C. 5 S C
Page 1 and 2: SIMPLE AND EXTENDED KALMAN FILTERS:
Page 3: There are different versions of Kal
Page 7 and 8: where : - r is the risk free intere
Page 9 and 10: In continuous time, the pricing equ
Page 11 and 12: empirical data are rebuilt. The rel
Page 13 and 14: The MPE is defined as follows: 1 MP
Page 15 and 16: with maturity, which is consistent
Page 17 and 18: Kalman filters are powerful tools s
Page 19 and 20: Lautier, D. (2000) La structure par
Page 21 and 22: ($/b) 7 6 5 4 3 2 1 0 -1 -2 -3 -4 -
Page 23 and 24: Table 1. Optimal parameters, 1995-1
Page 25 and 26: Table 3. The model’s performances
Page 27 and 28: Table 5. The simple filter with and
Page 29 and 30: Table 7. The performances with a 3
Page 31 and 32: APPENDIX: THE SIMPLE AND THE EXTEND
Page 33 and 34: where t / t −1 y represents an N
Page 35: 1 A more precise presentation of th

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